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\title{Derivations of two closely related algebras}
\author{LQW}

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\abstract{We study two closely related algebras and their spectra by the derivations on these algebras.}

\section{Problem}

Let  $Y=\text{Spec}(S)$ be the affine line where $S=k[t]$, Let $X=\text{Spec}(R)$ where $R=k[x,y]/(y^2-x^3)$. Consider two maps $t \mapsto (t^2,t^3)$ and $(x,y) \mapsto y/x$. Explain the relation between the derivations $\text{Der}(S)$ and $\text{Der}(R)$. Explain the relation between the tangent bundles $TY$ and $TX$. 





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